Flying in you'd want to fly to the Venice Area, either Venice's main airport, Marco Polo, or Treviso, or Bolzano in Italy. Or Innsbruck Austria.
Driving in is the most obvious option. Cortina is not too far from the Autobahn to the Brenner Pass or there is an Autostrada that goes from Venice to Cadore some 40km away from Cortina. Thankfully the valley does not have a very high elevation so it is clear of treacherous road conditions for much of the year, be sure to check road conditions before driving and be aware that conditions tend to be far worse through the mountain passes, be sure you are driving on the most traveled passes as they get cleared first.
Buses run somewhat frequently from Venice, but not entirely that often and the bus schedule seems to change frequently.
There is a train to Pieve Di Cadore some 35km from Cortina that comes direct from Venice (€7.50 each way) a 6 times a day during the peak winter season and takes approximately 2h45m, and then dolomitibus runs a bus that runs roughly 5-10min after arrival that will go to go to Cortina in approx a half hour, a ticket is €3 and can be purchased upon entering. Taxis to Cortina from here are €65. Similar public transport runs in summer